Stirling number

The Stirling numbers of the first and second kind, named after James Stirling , are used in combinatorics and theoretical computer science .

Designation and notation

With reference to a work by Stirling published as early as 1730, in which these numbers are examined, Niels Nielsen introduced the term "Stirling numbers of the first and second kind" in the handbook of the theory of the gamma function in 1906 ("nombres de Stirling" in an article published in 1904 ).

Neither the designation as Stirling numbers nor uniform notations have established themselves. In this article, Stirling numbers of the first kind are denoted with a lower case or written one above the other in square brackets, Stirling numbers of the second type are denoted with a capital letter or written above one another in curly brackets: ${\displaystyle s}$ ${\displaystyle S}$

${\displaystyle s_{n,k}=\left[{n \atop k}\right],\qquad S_{n,k}=\left\{{n \atop k}\right\}}$.

The bracket notation , also called Karamata notation , was introduced in 1935 by Jovan Karamata in analogy to the binomial coefficients , in 1992 Donald Knuth advocated this notation with a detailed excursus on the Stirling numbers. ${\displaystyle {\tbinom {n}{k}}}$

Stirling numbers of the first kind

The Stirling number of the first kind is the number of permutations of a -elemental set that have exactly cycles . According to another definition that is often used, it is instead referred to as the Stirling number of the first kind. ${\displaystyle s_{n,k}}$ ${\displaystyle n}$ ${\displaystyle k}$ ${\displaystyle (-1)^{n-k}s_{n,k}}$

example

The set with elements can be divided into cycles in the following ways : ${\displaystyle \{a,b,c,d\}}$ ${\displaystyle n=4}$ ${\displaystyle k=2}$

${\displaystyle (abc)(d),{\text{ }}(acb)(d),{\text{ }}(abd)(c),{\text{ }}(adb)(c),{\text{ }}(acd)(b),{\text{ }}(adc)(b),{\text{ }}(bcd)(a),{\text{ }}(bdc)(a),{\text{ }}(ab)(cd),{\text{ }}(ac)(bd),{\text{ }}(ad)(bc)}$

So is . For further examples see cycle type . ${\displaystyle s_{4,2}=11}$

properties

The explicit formulas apply

${\displaystyle s_{n,k}=\!\!\!\sum _{0<i_{1}<i_{2}<\ldots <i_{n-k}<n}\!\!\!i_{1}i_{2}\cdots i_{n-k}=(n-1)!\!\!\!\sum _{0<j_{1}<j_{2}<\ldots <j_{k-1}<n}\!\!\!(j_{1}j_{2}\cdots j_{k-1})^{-1}}$

and the recursive formula

${\displaystyle s_{n+1,k}=s_{n,k-1}+n\,s_{n,k}}$

with the initial conditions

${\displaystyle s_{n,n}=1}$     and

${\displaystyle s_{n,k}=0}$     for     or   ${\displaystyle k=0<n}$ ${\displaystyle n<k.}$

Other special values ​​are

• ${\displaystyle s_{n,1}=(n-1)!}$
• ${\displaystyle s_{n,2}=(n-1)!\,H_{n-1},}$Follow A000254 in OEIS
• ${\displaystyle s_{n,3}={\tfrac {1}{2}}\,(n-1)!\,(H_{n-1}^{2}-H_{n-1,2}),}$Follow A000399 in OEIS
• ${\displaystyle s_{n,n-3}={\tbinom {n}{4}}{\tbinom {n}{2}}={\tfrac {1}{48}}\,n^{2}\,(n-1)^{2}\,(n-2)\,(n-3),}$Follow A001303 in OEIS
• ${\displaystyle s_{n,n-2}={\tfrac {1}{24}}\,n\,(n-1)\,(n-2)\,(3n-1),}$Follow A000914 in OEIS
• ${\displaystyle s_{n,n-1}={\tbinom {n}{2}}={\tfrac {1}{2}}\,n\,(n-1)}$

all wherein the th harmonic number and is a generalized harmonic number. ${\displaystyle n>0,}$ ${\displaystyle H_{n-1}=1^{-1}+2^{-1}+...+(n-1)^{-1}}$ ${\displaystyle (n-1)}$ ${\displaystyle H_{n-1,2}=1^{-2}+2^{-2}+...+(n-1)^{-2}}$

In general, as a polynomial in the degree be considered. It has the leading coefficient and contains the factors n , n −1, ..., n - k for all and the factors n ² and ( n −1) ² for odd numbers . The polynomial ${\displaystyle s_{n,n-k}}$ ${\displaystyle n}$ ${\displaystyle 2k}$ ${\displaystyle 1/(2^{k}k!)}$ ${\displaystyle k>0}$ ${\displaystyle k>1}$

${\displaystyle \psi _{k}(n)=s_{n+1,n-k}/{\bigl (}(n+1)\,n\cdots (n-k){\bigr )}}$

in of degrees is also known as the Stirling polynomial , see also section Stirling polynomials . ${\displaystyle n}$ ${\displaystyle k}$

Generating functions are

${\displaystyle \sum _{n=0}^{\infty }s_{n,k}\,{\frac {t^{n}}{n!}}={\frac {1}{k!}}{\bigl (}\!-\log(1-t){\bigr )}^{k}}$     and         and ${\displaystyle \sum _{k=0}^{\infty }\sum _{n=0}^{\infty }s_{n,k}\,{\frac {t^{n}}{n!}}\,u^{k}=(1-t)^{-u}}$

${\displaystyle \sum _{k=0}^{n}s_{n,k}\,x^{k}=(x)^{n}}$

with the increasing factorial ${\displaystyle (x)^{n}=x\,(x+1)\,\cdots \,(x+n-1).}$

Is a prime number , then for by divisible and just by divisible ( Nielsen 1893). The Wolstenholme's theorem is the special case ${\displaystyle p}$ ${\displaystyle s_{p,k}}$ ${\displaystyle 1<k<p}$ ${\displaystyle p}$ ${\displaystyle k<p-1}$ ${\displaystyle p^{2}}$ ${\displaystyle k=2.}$

Since the number of all permutations is a -element set, it follows ${\displaystyle n!}$ ${\displaystyle n}$

${\displaystyle \sum _{k=0}^{n}s_{n,k}=n!}$

and in particular directly from the definition of ${\displaystyle s_{n,k}\leq n!}$ ${\displaystyle s_{n,k}.}$

For each there is such a that ${\displaystyle n>2}$ ${\displaystyle m_{n},}$

${\displaystyle s_{n,0}<s_{n,1}<\ldots <s_{n,m_{n}-1}<s_{n,m_{n}}>s_{n,m_{n}+1}>\ldots >s_{n,n}}$

and or ( Erdős 1953). ${\displaystyle m_{n+1}=m_{n}}$ ${\displaystyle m_{n+1}=m_{n}+1}$

For each , the sequence is strictly logarithmically concave , that is, for ${\displaystyle n}$ ${\displaystyle s_{n,0},s_{n,1},...,s_{n,n}}$ ${\displaystyle s_{n,k}^{2}>s_{n,k-1}s_{n,k+1}}$ ${\displaystyle 0<k<n.}$

The asymptotic behavior of assuming is ${\displaystyle s_{n,k}}$ ${\displaystyle k=o(\log n)}$

${\displaystyle s_{n,k}\sim {\frac {(n-1)!}{(k-1)!}}\,(\gamma +\log n)^{k-1}}$

with the Euler-Mascheroni constant ${\displaystyle \gamma .}$

Stirling numbers of the second kind

The Stirling number of the second kind is the number of -element partitions of a -element set, i.e. the number of possibilities to divide a -element set into non-empty disjoint subsets. ${\displaystyle S_{n,k}}$ ${\displaystyle k}$ ${\displaystyle n}$ ${\displaystyle n}$ ${\displaystyle k}$

${\displaystyle S_{n,k}}$is also the number of ways to divide distinguishable balls into indistinguishable compartments so that at least one ball is in each compartment. If the subjects are distinguishable, then one obtains possibilities; this is also the number of surjective mappings of a -element set to a -element set. ${\displaystyle n}$ ${\displaystyle k}$ ${\displaystyle k!S_{n,k}}$ ${\displaystyle n}$ ${\displaystyle k}$

example

The set with elements can be broken down into non-empty disjoint subsets in the following ways : ${\displaystyle \{a,b,c,d\}}$ ${\displaystyle n=4}$ ${\displaystyle k=2}$

${\displaystyle \{\{a,b\},\{c,d\}\},\ \{\{a,c\},\{b,d\}\},\ \{\{a,d\},\{b,c\}\},}$

${\displaystyle \{\{a,b,c\},\{d\}\},\ \{\{a,b,d\},\{c\}\},\ \{\{a,c,d\},\{b\}\},\ \{\{b,c,d\},\{a\}\}.}$

So is . ${\displaystyle S_{4,2}=7}$

properties

The explicit formulas apply

${\displaystyle S_{n,k}={\frac {1}{k!}}\sum _{j=0}^{k}(-1)^{k-j}{\binom {k}{j}}j^{\,n}}$     and

${\displaystyle S_{n,k}=\!\!\!\sum _{1\leq i_{1}\leq i_{2}\leq ...\leq i_{n-k}\leq k}\!\!\!i_{1}i_{2}\cdots i_{n-k}=\!\!\!\sum _{c_{1}+c_{2}+\cdots +c_{k}=n-k}\!\!\!1^{c_{1}}2^{c_{2}}\cdots k^{c_{k}}}$

with integer nonnegative and the recursive formula ${\displaystyle c_{1},c_{2},...,c_{k}}$

${\displaystyle S_{n+1,k}=S_{n,k-1}+k\,S_{n,k}}$

with the initial conditions

${\displaystyle S_{n,n}=1}$     and

${\displaystyle S_{n,k}=0}$     for     or   ${\displaystyle k=0<n}$ ${\displaystyle n<k.}$

Other special values ​​are

• ${\displaystyle S_{n,1}=1}$
• ${\displaystyle S_{n,2}=2^{n-1}-1}$
• ${\displaystyle S_{n,3}={\tfrac {1}{2}}\,(3^{n-1}-2^{n}+1),}$Follow A000392 in OEIS
• ${\displaystyle S_{n,n-3}={\tbinom {n}{4}}{\tbinom {n-2}{2}}={\tfrac {1}{48}}\,n\,(n-1)\,(n-2)^{2}\,(n-3)^{2},}$Follow A001297 in OEIS
• ${\displaystyle S_{n,n-2}={\tfrac {1}{24}}\,n\,(n-1)\,(n-2)\,(3n-5),}$Follow A001296 in OEIS
• ${\displaystyle S_{n,n-1}={\tbinom {n}{2}}={\tfrac {1}{2}}\,n\,(n-1)}$

for all ${\displaystyle n>0.}$

Can also be understood as a polynomial in degrees . It has the leading coefficient and contains the factors n , n −1,…, n - k for all and the factors ( n - k ) ² and ( n - k +1) ² for odd numbers . The same Stirling polynomial -th degree is obtained as with the Stirling numbers of the first kind using ${\displaystyle S_{n,n-k}}$ ${\displaystyle n}$ ${\displaystyle 2k}$ ${\displaystyle 1/(2^{k}k!)}$ ${\displaystyle k>0}$ ${\displaystyle k>1}$ ${\displaystyle k}$

${\displaystyle (-1)^{k}\psi _{k}(-n)=S_{n+k,n-1}/{\bigl (}(n+k)\,(n+k-1)\cdots (n-1){\bigr )}.}$

Generating functions are

${\displaystyle \sum _{n=0}^{\infty }S_{n,k}\,{\frac {t^{n}}{n!}}={\frac {1}{k!}}(e^{t}-1)^{k}}$     and         and ${\displaystyle \sum _{k=0}^{\infty }\sum _{n=0}^{\infty }S_{n,k}\,{\frac {t^{n}}{n!}}\,u^{k}=e^{(e^{t}-1)u}}$

${\displaystyle \sum _{n=0}^{\infty }S_{n,k}\,u^{n}={\frac {u^{k}}{(1-u)(1-2u)\cdots (1-ku)}}}$     and

${\displaystyle \sum _{k=0}^{n}S_{n,k}\,(x)_{k}=x^{n}}$

with the falling factorial ${\displaystyle (x)_{n}=x\,(x-1)\,\cdots \,(x-n+1).}$

Is a prime number, then for is divisible by. ${\displaystyle p}$ ${\displaystyle S_{p,k}}$ ${\displaystyle 1<k<p}$ ${\displaystyle p}$

Since Bell's number is the number of all partitions in a -element set, the following applies ${\displaystyle B_{n}}$ ${\displaystyle n}$

${\displaystyle \sum _{k=0}^{n}S_{n,k}=B_{n}.}$

The Bernoulli number β _n is obtained as the alternating sum

${\displaystyle \sum _{k=0}^{n}(-1)^{k}{\frac {k!}{k+1}}S_{n,k}=\beta _{n}.}$

With the help of the recursion formula one can show that for every one exists such that ${\displaystyle n>2}$ ${\displaystyle m_{n}}$

${\displaystyle S_{n,0}<S_{n,1}<\ldots <S_{n,m_{n}-1}\leq S_{n,m_{n}}>S_{n,m_{n}+1}>\ldots >S_{n,n}}$

and or applies. It is an open question whether there exists one for which the case occurs. ${\displaystyle m_{n+1}=m_{n}}$ ${\displaystyle m_{n+1}=m_{n}+1}$ ${\displaystyle n>2}$ ${\displaystyle S_{n,m_{n}-1}=S_{n,m_{n}}}$

For each , the sequence is strictly logarithmically concave , that is, for ${\displaystyle n}$ ${\displaystyle S_{n,0},S_{n,1},...,S_{n,n}}$ ${\displaystyle S_{n,k}^{2}>S_{n,k-1}S_{n,k+1}}$ ${\displaystyle 0<k<n.}$

Relationship between the Stirling numbers of the first and second kind

From relationships

${\displaystyle x^{n}=\sum _{k=0}^{n}S_{n,k}\,(x)_{k}}$     and     ${\displaystyle (x)_{n}=\sum _{k=0}^{n}(-1)^{n-k}s_{n,k}\,x^{k},}$

which are also often used to define the Stirling numbers of the second and first kind, it follows that these are the coefficients of mutually inverse linear transformations, the Stirling transformation and the inverse Stirling transformation . This means that the lower triangular matrices and are mutually inverse matrices : ${\displaystyle {\bigl (}S_{n,k}{\bigr )}_{n,k}}$ ${\displaystyle {\bigl (}(-1)^{n-k}s_{n,k}{\bigr )}_{n,k}}$

${\displaystyle \sum _{i=0}^{\infty }S_{n,i}\,(-1)^{i-k}s_{i,k}=\delta _{n,k}=\sum _{i=0}^{\infty }(-1)^{n-i}s_{n,i}\,S_{i,k}}$

with the Kronecker delta for and for ${\displaystyle \delta _{n,k}=1}$ ${\displaystyle n=k}$ ${\displaystyle \delta _{n,k}=0}$ ${\displaystyle n\neq k.}$

The Stirling numbers of the first and second kind can each be represented by the other ( Schlömilch 1852):

${\displaystyle (-1)^{n-k}s_{n,k}=\sum _{j=0}^{n-k}(-1)^{j}{\binom {n+j-1}{k-1}}{\binom {2n-k}{n-k-j}}S_{n-k+j,j}}$     and

${\displaystyle S_{n,k}=\sum _{j=0}^{n-k}(-1)^{j}{\binom {n+j-1}{k-1}}{\binom {2n-k}{n-k-j}}(-1)^{n-k}s_{n-k+j,j}.}$

The Stirling numbers can clearly be continued on negative whole indices and that the recursion formulas ${\displaystyle n}$ ${\displaystyle k}$

${\displaystyle s_{n+1,k}=s_{n,k-1}+n\,s_{n,k}}$     and     ${\displaystyle S_{n+1,k}=S_{n,k-1}+k\,S_{n,k}}$

apply in general and for One receives the duality valid for all whole numbers and ${\displaystyle S_{n,k}=0=s_{n,k}}$ ${\displaystyle n<k=0.}$ ${\displaystyle n}$ ${\displaystyle k}$

${\displaystyle S_{-n,-k}=s_{k,n}\,,}$

the recursion formulas, the two into one another, also for one as polynomials in Sets in the conceived and for negative whole numbers, one obtains the same Continued negative integer indices and the polynomials for the duality ${\displaystyle S_{n,k}=0=s_{n,k}}$ ${\displaystyle n{\text{ }}k<0.}$ ${\displaystyle n}$ ${\displaystyle S_{n,n-k}}$ ${\displaystyle s_{n,n-k}}$ ${\displaystyle n}$

${\displaystyle S_{n,n-k}=s_{(k-n),(k-n)-k}\,.}$

Analogy to the binomial coefficients

The following applies to the binomial coefficients

${\displaystyle {\binom {n+1}{k}}={\binom {n}{k-1}}+{\binom {n}{k}}.}$

The Karamata notation emphasizes the analogy:

${\displaystyle {\Bigl [}{n+1 \atop k}{\Bigr ]}={\Bigl [}{n \atop k-1}{\Bigr ]}+n\,{\Bigl [}{n \atop k}{\Bigr ]}}$

${\displaystyle {\Bigl \{}{n+1 \atop k}{\Bigr \}}={\Bigl \{}{n \atop k-1}{\Bigr \}}+k\,{\Bigl \{}{n \atop k}{\Bigr \}}}$

Accordingly, the Stirling numbers can be arranged in a triangle scheme similar to Pascal's triangle and calculated line by line.

Triangle for Stirling numbers of the first kind (first line, first column, sequence A130534 in OEIS ): ${\displaystyle n=1,}$ ${\displaystyle k=1;}$

                             1
                          1     1
                       2     3     1
                    6    11     6     1
                24    50    35    10     1
             120   274   225   85    15     1
          720  1764  1624   735   175   21     1
      5040  13068 13132 6769  1960   322   28     1
  40320 109584 118124 67284 22449 4536  546   36     1
...   ...    ...   ...   ...   ...   ...   ...   ...    1

Triangle for Stirling numbers of the second kind (first line, first column, sequence A008277 in OEIS ): ${\displaystyle n=1,}$ ${\displaystyle k=1;}$

                             1
                          1     1
                       1     3     1
                    1     7     6     1
                 1    15    25    10     1
              1    31    90    65    15     1
           1    63    301   350   140   21     1
        1    127   966  1701  1050   266   28     1
     1    255  3025  7770  6951  2646  462    36     1
  1    ...   ...   ...   ...   ...   ...   ...   ...    1

As a further analogy, there are injective and surjective functions with -elementary definition and -elementary target sets . ${\displaystyle k!{\tbinom {n}{k}}}$ ${\displaystyle \textstyle n!{\bigl \{}\!{k \atop n}\!{\bigr \}}}$ ${\displaystyle k}$ ${\displaystyle n}$

Stirling polynomials

The Stirling polynomials introduced in the section Stirling numbers of the first kind are also used by the generating functions

${\displaystyle {\Bigl (}{\frac {t}{1-e^{-t}}}{\Bigr )}^{x+1}=1+(x+1)\sum _{k=0}^{\infty }\psi _{k}(x)\,t^{k+1}}$     and

${\displaystyle {\Bigl (}{\frac {-\log(1-t)}{t}}{\Bigr )}^{x}=1+x\sum _{k=0}^{\infty }\psi _{k}(x+k)\,t^{k+1}}$

which are obtained by generalizing generating functions of and . According to another definition, the polynomials and are called Stirling polynomials. The polynomials ψ ₀ ( x ), ψ ₁ ( x ),…, ψ ₆ ( x ) are ${\displaystyle S_{n,k}}$ ${\displaystyle s_{n,k}}$ ${\displaystyle S_{0}(x)=1}$ ${\displaystyle S_{k}(x)=k!(x+1)\psi _{k-1}(x)}$

${\displaystyle {\tfrac {1}{2}},}$     ${\displaystyle {\tfrac {1}{24}}(3x+2),}$     ${\displaystyle {\tfrac {1}{48}}(x+1)\,x,}$     ${\displaystyle {\tfrac {1}{5760}}(15x^{3}+15x^{2}-10x-8),}$     ${\displaystyle {\tfrac {1}{11520}}(x+1)\,x\,(3x^{2}-x-6),}$

${\displaystyle {\tfrac {1}{2903040}}(63x^{5}-315x^{3}-224x^{2}+140x+96),}$     ${\displaystyle {\tfrac {1}{5806080}}(x+1)\,x\,(9x^{4}-18x^{3}-57x^{2}+34x+80),}$

and special values ​​for are ${\displaystyle k>0}$

${\displaystyle \psi _{k}(-1)=-\beta _{k+1}/((k+1)!(k+1))}$     and     ${\displaystyle \psi _{k}(0)=\beta _{k+1}/(k+1)!}$

with the Bernoulli number β _{k +1} . The polynomials can be calculated with the formulas

${\displaystyle \psi _{k}(x)=\sum _{j=0}^{k}{\frac {C_{k+1,j}}{(2k+2-j)!}}(x-k-1)_{k-j}}$     and

${\displaystyle (-1)^{k}\psi _{k}(-x)=\sum _{j=0}^{k}{\frac {{\overline {C}}_{k+1,j}}{(2k+2-j)!}}(x-2)_{k-j}}$

with the through for j ∉ {0, 1,…, k −1} and ${\displaystyle C_{1,0}=1,}$ ${\displaystyle C_{k,j}=0}$

${\displaystyle C_{k+1,j}=(2k+1-j)\,(C_{k,j-1}+C_{k,j}),}$see episode A111999 in OEIS ,

and that by C̅ _1,0 = 1, C̅ _{k , j} = 0 for j ∉ {0, 1,…, k −1} and

${\displaystyle {\overline {C}}_{k+1,j}=(k+1-j)\,{\overline {C}}_{k,j-1}+(2k+1-j)\,{\overline {C}}_{k,j}}$

recursively defined integer coefficients. For one receives ${\displaystyle k>0}$

${\displaystyle s_{n,n-k}=\sum _{j=0}^{k-1}C_{k,j}{\binom {n}{2k-j}}}$     and     ${\displaystyle S_{n,n-k}=\sum _{j=0}^{k-1}{\overline {C}}_{k,j}{\binom {n}{2k-j}}.}$

This calculation of and is particularly efficient for large and small . ${\displaystyle s_{n,n-k}}$ ${\displaystyle S_{n,n-k}}$ ${\displaystyle n}$ ${\displaystyle k}$

literature

• Niels Nielsen : Faculties and faculty coefficients , Chapter 5 in the manual of the theory of the gamma function , BG Teubner, Leipzig 1906, pp. 66–78 (conversion and ; in the Internet archive , ditto , ditto ; yearbook report ) ${\displaystyle C_{n}^{p}=s_{n,n-p}}$ ${\displaystyle {\mathfrak {C}}_{\,n}^{\,p}=S_{n-1+p,n-1}}$
• Leonard Eugene Dickson : History of the theory of numbers. Volume I: Divisibility and primality , Carnegie Institution, Washington 1919, especially pp. 95-103 (English; conversion and ; in the Internet archive ; yearbook report ) ${\displaystyle S_{n,m}=s_{m+1,m+1-n}}$ ${\displaystyle \sigma _{n,m}=S_{m+n,m}}$
• Károly Jordan (Charles Jordan): Stirling's numbers , Chapter 4 in Calculus of finite differences , Chelsea, Budapest 1939, 2nd edition New York 1947 1960, 3rd edition 1965 1979, ISBN 0-8284-0033-4 , p. 142– 229 (English; conversion and ) ${\displaystyle S_{n}^{m}=(-1)^{n-m}s_{n,m}}$ ${\displaystyle {\mathfrak {S}}_{n}^{m}=S_{n,m}}$
• Milton Abramowitz , Irene A. Stegun (Ed.): Handbook of mathematical functions with formulas, graphs, and mathematical tables (10th edition), US Department of Commerce / National Bureau of Standards, 1972, pp. 822, 824-825, 833-835 (English; at ConvertIt.com ; at SFU Burnaby ; Zentralblatt review )
• Louis Comtet : Advanced combinatorics: the art of finite and infinite expansions , D. Reidel, Dordrecht 1974, ISBN 90-277-0441-4 , pp. 204–229 (English)
• Martin Aigner : Combinatorial Theory. Springer, Berlin 1997, ISBN 3-540-61787-6 (English; new edition of the 1979 edition; Zentralblatt review )
• Ronald L. Graham , Donald E. Knuth , Oren Patashnik : Concrete mathematics: a foundation for computer science , Addison-Wesley, Reading 1988, 2nd edition 1994, ISBN 0-201-55802-5 , pp. 257-266 (English ; Knuth's website for the book with errata: Concrete Mathematics, Second Edition ; Zentralblatt review )

Individual evidence

1. ↑ James Stirling : Methodus Differentialis: sive Tractatus de Summatione et Interpolatione Serierum Infinitarum , G. Strahan, Londini (London) 1730 (Latin; table of the Stirling numbers of the second kind on p. 8 , the Stirling numbers of the first kind on p. 11 )
2. ^ Nielsen: Faculties and faculty coefficients , 1906, pp. 66–67
3. ↑ Niels Nielsen : Recherches sur les polynomes et les nombres de Stirling , Annali di matematica pura ed applicata 10, 1904, pp. 287-318 (French)
4. ^ Henry W. Gould : Once again the Stirling numbers , annual report of DMV 73, 1971/72, pp. 149–152
5. ↑ ^{a b} Donald E. Knuth : Two notes on notation , The American Mathematical Monthly 99, 1992, pp. 403-422 (English; Zentralblatt review )
6. ↑ Jovan Karamata : Théorèmes sur la sommabilité exponentielle et d'autres sommabilités s'y rattachant (May 21, 1932), Mathematica (Cluj) 9, 1935, pp. 164–178 (French; Zentralblatt review )
7. ↑ ^{a b} Nielsen: Faculties and faculty coefficients , 1906, p. 72 ff.
8. ↑ Comtet: Advanced combinatorics , 1974, p. 218
9. ↑ Niels Nielsen: Om Potenssummer af hele Tal , Nyt Tidsskrift for Mathematik B 4, 1893, pp. 1–10 (Danish; Formula 17 on p. 4 with ; yearbook report ) ${\displaystyle A_{n,p}=s_{n+1,n+1-p}}$
10. ↑ Paul Erdős : On a conjecture of Hammersley , Journal of the London Mathematical Society 28, 1953, pp. 232-236 (English; only the proof for is not elementary; Zentralblatt review ) ${\displaystyle s_{n,m_{n}-1}\neq s_{n,m_{n}}}$
11. ^ ^{A b} Elliott H. Lieb : Concavity properties and a generating function for Stirling numbers , Journal of Combinatorial Theory 5, September 1968, pp. 203-206 (English; Zentralblatt review )
12. ↑ Comtet: Advanced combinatorics , 1974, p. 219
13. ^ E. Rodney Canfield, Carl Pomerance : On the problem of uniqueness for the maximum Stirling number (s) of the second kind , Integers 2, 2002, A01 (English; corrigendum ; Zentralblatt review )
14. ↑ Oskar Schlömilch : Recherches sur les coefficients des facultés analytiques , Journal for pure and applied mathematics 44, 1852, pp. 344–355 (French; Formula 14 on p. 346 with and ) ${\displaystyle C_{k}^{+n}=s_{n,n-k}}$ ${\displaystyle C_{k}^{-n}=S_{n+k,n}}$
15. ↑ Ira Gessel, Richard P. Stanley : Stirling polynomials ( PDF file, 534 kB), Journal of combinatorial theory A 24, 1978, pp. 24–33 (English; Zentralblatt review )
16. ↑ Antal E. Fekete : Apropos Two notes on notation , The American Mathematical Monthly 101, October 1994, pp. 771-778 (English; Zentralblatt review )
17. ↑ Jordan: Stirling's numbers , 1979, pp. 147-153
18. ↑ Jordan: Stirling's numbers , 1979, pp. 168-173

Web links

Wikiversity: A lecture on partitions and Stirling numbers of the second kind as part of a course on discrete mathematics  - course materials

• Eric W. Weisstein: Stirling Number of the First Kind , Stirling Number of the Second Kind , Stirling Transform and Stirling Polynomial . In: MathWorld . (English)
• Stirling number of the first kind and Stirling number of the second kind at The Wolfram Functions Site (English; with calculation option)
• Set Partitions: Stirling Numbers in the NIST Digital Library of Mathematical Functions (English)